Lemma 63.11.7 (0GLE)—The Stacks project

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Table of contents
Part 3: Topics in Scheme Theory
Chapter 63: More Étale Cohomology
Section 63.11: Derived upper shriek
Lemma 63.11.7 (cite)

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Lemma 63.11.7. Consider a cartesian square

$\xymatrix{ X' \ar[r]_{g'} \ar[d]_{f'} & X \ar[d]^ f \\ Y' \ar[r]^ g & Y }$

of quasi-compact and quasi-separated schemes with $f$ separated and of finite type. Then we have $Rf^! \circ Rg_* = Rg'_* \circ R(f')^!$ .

Proof. By uniqueness of adjoint functors this follows from base change for derived lower shriek: we have $g^{-1} \circ Rf_! = Rf'_! \circ (g')^{-1}$ by Lemma 63.9.4. $\square$

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